Select the correct answer.

Nancy is given a problem in her math class. In order to solve the problem, she must find the product of two numbers. The first number is [tex]\(14 - x\)[/tex], and the second number is [tex]\(19 + 7x\)[/tex].

Which of the following functions represents the product of the two numbers?

A. [tex]\(p(x) = 7x^2 - 215x + 1,638\)[/tex]
B. [tex]\(p(x) = 7x^2 - 117x + 686\)[/tex]
C. [tex]\(p(x) = 7x^2 - 98x + 819\)[/tex]
D. [tex]\(p(x) = 7x^2 - 117x + 819\)[/tex]

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Answer :

To solve this problem, let’s break it down step-by-step.

1. Identify the expressions for the numbers:
- The first number is [tex]\( 14 - x \)[/tex].
- The second number is 19 more than 7 times the first number. Mathematically, this can be expressed as [tex]\( 19 + 7(14 - x) \)[/tex].

2. Simplify the expression for the second number:
[tex]\[ 19 + 7(14 - x) = 19 + 98 - 7x = 117 - 7x \][/tex]
Therefore, the second number is [tex]\( 117 - 7x \)[/tex].

3. Calculate the product of the two numbers:
[tex]\[ (14 - x)(117 - 7x) \][/tex]

Now, we need to expand this multiplication:
[tex]\[ (14 - x)(117 - 7x) = 14 \cdot 117 + 14 \cdot (-7x) + (-x) \cdot 117 + (-x) \cdot (-7x) \][/tex]
[tex]\[ = 1638 - 98x - 117x + 7x^2 \][/tex]
Combine the like terms:
[tex]\[ = 7x^2 - 215x + 1638 \][/tex]

4. Compare with the given functions:

We need to identify the function that matches the simplified expression [tex]\( 7x^2 - 215x + 1638 \)[/tex].

- [tex]\( p(x) = 7x^2 - 215x + 1638 \)[/tex]
- [tex]\( p(x) = 7x^2 - 117x + 686 \)[/tex]
- [tex]\( p(x) = 7x^2 - 98x + 819 \)[/tex]
- [tex]\( p(x) = 7x^2 - 117x + 819 \)[/tex]

The correct function is [tex]\( p(x) = 7x^2 - 215x + 1638 \)[/tex], but since this is not among the given options, the actual correct function should match the simplified product:

Thus, the correct choice should be:
[tex]\( \boxed{7 x^2-117 x+819} \)[/tex]